The odd numbers lying between 100 and 200 are
101, 103, 105,…, 199
a2 – a1 = 103 – 101 = 2
a3 – a2 = 105 – 103 = 2
∵ a3 – a2 = a2 – a1 = 2
Therefore, the series is in AP
Here, a = 101, d = 2 and an = 199
We know that,
an = a + (n – 1)d
⇒ 199 = 101 + (n – 1)2
⇒ 199 – 101 = (n – 1)2
⇒ 98 = (n – 1)2
⇒ 49 = (n – 1)
⇒ n = 50
Now, we have to find the sum of this AP
⇒ S50 = 25[202 + 49 × 2]
⇒ S50 = 25[300]
⇒ S50 = 7500
Hence, the sum of all odd numbers lying between 100 and 200 is 7500.